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CIGRE US National Committee
2019 Grid of the Future Symposium
An Optimization-based Method in Determining the Capability Curve of a
Microgrid
S. HASAN, A. MAJZOOBI,
A. KHODAEI
University of Denver
USA
C. HENG, L. ZHANG
ComEd
USA
SUMMARY
The capability curve of generators is essential in power system operation as it determines a
generator’s capability in delivering real and reactive powers. Reactive power dispatch is an
integral part of power system operation to ensure system load balance and further manage
voltage stability. Although these capability curves are provided by generator’s manufacturers
and readily available for power system studies, the same is not true for microgrids which
comprise various distributed energy resources. The main purpose of this paper is to find the
capability curve of a microgrid as a single unit (i.e., a virtual power plant) in a distribution grid.
Obtaining the capability curve of the microgrid provides the microgrid owner with a better
understanding on how much ancillary services the microgrid can offer to the utility grid, mainly
via reactive power production. To determine the microgrid capability curve, an optimizationbased method that identifies the maximum and minimum reactive power capability in various
real power operation points is proposed. The results of numerical analyses in this paper show
how the reactive power behaves under changing of real power production in a microgrid and
how these two outputs are correlated.
KEYWORDS
Ancillary services, capability curve, distributed energy resource, microgrid, reactive power.
Amin.Khodaei@du.edu
1. INTRODUCTION
A microgrid is a small scale power grid, operated at low voltage level, which is implemented
by integrating various distributed energy resources (DERs) with the goal of improving local
reliability, resilience, and power quality [1]-[3]. In addition to all merits of microgrids,
microgrids have also been considered as viable ancillary service providers to utility grids and
by increasing their penetration in the network, this role is now ever more attractive [4]-[6].
Considering that all common microgrid DERs can produce reactive power (diesel generators
and microturbine through their synchronous generator and solar PV, wind, and batteries through
their power electronics interface), microgrids can potentially provide reactive power support to
the utility grid in terms of both injecting or withdrawing reactive power.
The reactive power management and voltage stability are strongly related to each other to keep
voltage in its proper operational limits. Reactive power requirements usually change over time
as loads change and accordingly inductors are used to manage high voltage on transmission
systems, while capacitors are used to control low voltage on both transmission and distribution
systems. In [7], an optimal reactive power dispatch is proposed to improve voltage stability
margin and increase reactive power reserve. Droop control is introduced to adjust reactive
power in order to keep voltage stability in microgrids [8,9].
In order to provide the ancillary services to the utility grid, the amount of exchanged real and
reactive power between microgrid and the utility grid should be managed. Therefore, obtaining
the microgrid power exchange capability, both real and reactive power, is important. The
capability curve of DERs could be defined as a boundary within which the generator operates
safely. Accordingly, reaching to microgrid capability curve is the first and most important step
for microgrid owners in order to contribute to ancillary service needs of the grid. To this end,
the capability curve of all DERs in microgrid should be combined to achieve an integrate
capability curve for the microgrid.
In synchronous generators, the P-Q capability curve is ruled by some constrains that are
explained and calculated in [10]. Leveraging solar PV as a reactive power source and
determining its limits and capacity is described in [11]. The possibility of using solar PV as a
source of ancillary services to support reactive power compensation, as well as modeling an
approach to develop capability curve of solar PV using an advanced control method is proposed
in [12]. Real and reactive power in solar PVs could be controlled by various algorithms,
applicable to fixed real power mode and maximum power point tracking (MPPT) control mode,
that provide fast reactive power with voltage stability during switching between these two
modes as discussed in [13]. Energy storage systems could also be used to improve the power
quality, including frequency and voltage level and support the system with needed backup
power [14]. The battery energy storage system would help in mitigating power fluctuations of
PV units and wind turbines under various weather conditions [15]. During the daytime, the
battery gets charged from PV system, and during the nighttime the battery discharges to the
critical loads or feeds the grid for reactive power and harmonic compensations. A reactive
power controller is designed in [16] to control the amount of reactive power delivered to the
grid. This controller operates based on power factor measurement.
The objective of this paper is to determine the real-reactive power capability curve of a
microgrid with various DERs. To achieve such a curve, the reactive power profile at different
load points is determined using a proposed optimization-based method, and consequently the
capability curve is drawn. By obtaining capability curve of the microgrid, the microgrid with
1
different DERs is seen as a single unit in the network, or a virtual power plant, with the ability
to offer reactive power. The rest of the paper is organized as follows. In Section 2, the limits of
reactive power for each individual DER is explained briefly. In Section 3, the capability curve
of a microgrid is calculated using the proposed optimization-based method. Section 4 concludes
the paper.
2. CAPABILITY CURVE OF INDIVIDUAL DERS
The capability curve of a microgrid is determined based on the individual capability curve of
each of its DERs. The discussion on the capability curve of synchronous generator, solar PV,
and battery energy storage is provided below.
Synchronous generator: The capability curve of a synchronous generator is considered as a
semicircle with radius S as apparent power, as shown in Fig. 1. The relation between real power
(P) and reactive power (Q) is shown in (1) and the limits of apparent (S), real, and reactive
powers are illustrated in (2)-(4), respectively.
𝑆 2 = 𝑃2 + 𝑄 2
0 ≤ 𝑆 ≤ 𝑆 𝑚𝑎𝑥
0 ≤ 𝑃 ≤ 𝑃𝑚𝑎𝑥
𝑄 𝑚𝑖𝑛 ≤ 𝑄 ≤ 𝑄 𝑚𝑎𝑥
Reactive Power (VAR)
(1)
(2)
(3)
(4)
Real Power (W)
Figure 1: Synchronous Generator and Solar PV capability curve.
Solar PV: The PV unit contains three elements: PV panels, inverters, and transformers [17].
The PV generator is controlled to operate at MPPT and the PV array is modeled as a current
source connected in parallel with a capacitor, while the inverter is modeled as a voltage source.
The PV unit’s output power depends on weather conditions and solar radiation during the day.
Maximum delivered real power by PV unit occurs at maximum solar radiation and minimum
temperature [18]. The DC voltage of solar PV inverters may limit the reactive power capability
of the inverters. The reactive power capability of clustered inverters in PV units is analyzed in
[19]. The relationships between apparent, real, and reactive powers are as those in (1)-(4) for a
solar PV unit, and the capability curve is shown in Fig. 1, the same as for a synchronous
generator.
Battery energy storage: The battery energy storage is used to seamlessly supply the loads
during peak hours and to capture power fluctuations of variable DERs. The battery energy
storage system has two operating modes of charging and discharging, in which its power is
positive at discharging mode and negative at charging mode. The battery energy storage
2
system’s capability curve is a circle with radius S, and with both leading and lagging power
factors associated with its operating modes, as shown in Fig. 2. The relation between real power
(P) and reactive power (Q) of battery energy storage system is shown in (5) and the limits of
apparent power (S), real power (P), and reactive power (S) are illustrated in (6)-(9), respectively.
𝑆 2 = 𝑃2 + 𝑄 2
𝑆 𝑚𝑖𝑛 ≤ 𝑆 ≤ 𝑆 𝑚𝑎𝑥
𝑃𝑚𝑖𝑛 ≤ 𝑃 ≤ 𝑃𝑚𝑎𝑥
𝑄 𝑚𝑖𝑛 ≤ 𝑄 ≤ 𝑄 𝑚𝑎𝑥
Reactive Power (VAR)
(5)
(6)
(7)
(8)
Real Power (W)
Figure 2: Battery energy storage capability curve.
3. MODEL FORMULATION
To find the microgrid capability curve, an optimization model is proposed as follows:
min/ max 𝑄
𝑠. 𝑡.
𝑃𝑠 + 𝑃𝑏 + 𝑃𝑔 = 𝑃𝑑
and (1) − (8)
(9)
(10)
(11)
The objective of this model is to find the minimum and the maximum reactive power that can
be produced by the microgrid (9) based on the net real power it produces. This objective is
subject to a load balance constraint (10) in which the summation of power output of solar PV
(Ps), battery (Pb) and the synchronous generator (Pg) equals a hypothetical net load (Pd). The
objective is further subject to constraints associated with individual DERs’ capability curves as
in (11).
Variables Ps and Pg are positive, while Pb can be either positive or negative based on the
battery’s charging and discharging state. Pd is initially assumed to be equal to the maximum
charging power of the battery (a negative value) and then in each several iterations Pd is
increased by a selected step. In each iteration for the selected Pd value, minimum and maximum
amount of reactive power that can be produced (two points) are calculated and then Pd is
increased by the selected step. This process continues until Pd reaches the generation capacity
of the microgrid. In that point the reactive power generation will reach zero as all DERs will
generate maximum real power and therefore their respective reactive power generations will be
zero, i.e., intersection of their capability curves with the horizontal real power axis.
3
Given the small size of this optimization problem, a very small step can be considered and the
problem can be solved from several iterations to find a very accurate capability curve. The
proposed model is generic and can consider any number of DERs as long as individual
capability curves are available.
4. NUMERICAL ANALYSES
A microgrid consisting of a synchronous generator, a solar PV, and an energy storage is
considered. In order to cover various conditions, three cases with various power capacities for
microgrid DERs are considered as follows:
Case 1: All three DERs have the same capacity.
Case 2: Two DERs (PV and energy storage) have the same capacity while the synchronous
generator is larger.
Case 3: The DERs have different capacities.
The capacities of microgrid components in aforementioned cases are tabulated in Table 1.
Table 1 Microgrid DERs Capacities
Case Number
Case 1
Case 2
Case 3
Synchronous
generator (MVA)
10
20
15
Solar Photovoltaic
(MVA)
10
10
5
Energy Storage
(MVA)
10
10
10
The proposed model is applied to these cases and the obtained results are plotted in Figs. 3-5.
As the figures show, capability curve of microgrid in all three cases are different from capability
curve of each component of the microgrid, i.e. semicircle for synchronous generator and solar
PV, and full circle for energy storage. Moreover, the figures clearly prove that the capability
curve of the microgrid depends on the capacities of its DERs.
Capability curve of microgrid in all three cases are non-symmetrical, however, Case 1 (Fig. 3)
has the closest curve to capability curve of each DER since the capacity of all components are
the same. Capability curve of microgrid in Case 2 (Fig. 4) has a breakpoint, stemming from the
different capacity of the synchronous generator. Fig. 5 depicts the most non-symmetrical figure,
belonging to Case 3, with two breakpoints in the curve which stems from the difference in
capacities. Thus, the results demonstrate that the shape of microgrid capability curve is affected
by the size of units in the microgrid for each case.
It is interesting to note that the shape of the microgrid capability curve is much different from
individual DERs; first the shape is not a circle or half circle as is common in DERs; second it
has a negative part as well which is resulted from battery charging. However similar to
individual DERs’ capability curves, once microgrid’s capability curve is obtained, a closed
form mathematical model can be fitted into the curve to be used for ancillary service studies.
4
Reactive Power (Mvar)
-15
-10
35
30
25
20
15
10
5
0
-5 -5 0
-10
-15
-20
-25
-30
-35
5
10
15
20
25
30
35
Active Power (MW)
Figure 3: Capability curve of microgrid for Case1.
50
Reactive Power (Mvar)
40
30
20
10
0
-20
-10
-10
0
10
20
30
40
50
-20
-30
-40
-50
Active Power (MW)
Figure 4: Capability curve of microgrid for Case 2.
40
Ractive Power (Mvar)
30
20
10
0
-15
-10
-5
0
5
10
15
20
25
30
35
-10
-20
-30
-40
Active Power (MW)
Figure 5: Capability curve of microgrid for Case 3.
5
5. CONCLUSIONS
Microgrids can be utilized as ancillary service providers in the distribution grid, besides all their
other benefits. Accordingly, capability curve of microgrid plays a key role in participating in
ancillary service market. This paper developed an optimization-based model to find the
capability curve of microgrid as a single unit. The numerical analyses were carried out in
various cases and the obtained results showed that the capability curve of microgrid depends
on the capacity of its DERs. The shape of capability curve would be significantly changed when
the DERs have different capacities.
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